# Numerical integrating regions to avoid oscillations

Lets define the functions

osc = E^(-((I (4 x^2 - 4 y^2))/(4 \[Pi])));
const = 1/4  Sech[1/2 (-x - y)]^2 Sech[(x - y)/2]^2 Sinh[x]^2;


the const function goes to 1 as either Abs[x]>Abs[y] go to infinity, and is exponentially small in the other regime. while the osc function is just an oscillating gaussian. I want to compute the integral

int=NIntegrate[ osc const, {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]}]


I know that in the Abs[x]>Abs[y] regime, this integral should essentially be

Integrate[osc, {x, -\[Infinity], \[Infinity]}, {y, -\[Infinity], \[Infinity]}]


which evaluates to Pi^2. while for Abs[x]<Abs[y] I should just get some finite contribution. However, when I try to simply compute int using Nintegrate as above, I am told that the integral does not converge, which is clearly wrong. How can I get the finite answer I expect?

• i am explaining that in the regime Abs[x]>Abs[y], you can just set const=1 to good accuracy, so that the integral can be done analytically. i agree that its not exactly as i wrote, since the range of integration also needs to be restricted (i just wrote that for simplicity) – esches Oct 28 '20 at 13:10
• i will edit it for clarity – esches Oct 28 '20 at 13:10